On regularity properties of extremal controls

We prove some regularity properties of the optimal controls for the smooth bracket generating systems with scalar control parameters, and show that the Cantor sets cannot be the sets of switching points.Partially supported by the Russian Fund for Fundamental Research and grant No. 95-01-00310a.     

The dynamic interpolation problem: On Riemannian manifolds,Lie groups,and symmetric spaces

We consider the dynamic interpolation problem for nonlinear control systems modeled by second-order differential equations whose configuration space is a Riemannian manifoldM. In this problem we are given an ordered set of points inM and would like to generate a trajectory of the system through the application of suitable control functions, so that the resulting trajectory in configuration space interpolates the given set of points. We also impose smoothness constraints on the trajectory and typically ask that the trajectory be also optimal with respect to some physically interesting cost function. Here we are interested in the situation where the trajectory is twice continuously differentiable and the Lagrangian in the optimization problem is given by the norm squared acceleration along the trajectory. The special cases whereM is a connected and compact Lie group or a homogeneous symmetric space are studied in more detail.Work supported in part by NSF grant No. 89-14643 and NATO project CRG 910926.     

On mappings of bounded variation

We present the properties of mappings of bounded variation defined on a subset of the real line with values in metric and normed spaces and show that major aspects of the theory of realvalued functions of bounded variation remains valid in this case. In particular, we prove the structure theorem and obtain the continuity properties of these mappings as well as jump formulas for the variation. We establish the existence of Lipschitz continuous geodesic paths and prove an analog of the well-known Helly selection principle. For normed space-valued smooth mappings we obtain the usual integral formula for the variation without the completeness assumption on the space of values. As an application of our theory we show that compact set-valued mappings (=multifunctions) of bounded variation admit regular selections of bounded variation. Partially supported by the Russian Foundation for Fundamental Research, Grant No. 96-01-00278.     

A survey of singular curves in sub-Riemannian geometry

Sub-Riemannian geometry is the geometry of a distribution ofk-planes on an-dimensional manifold with a smoothly varying inner product on thek-planes. Singular curves are singularities of the space of paths tangent to the distribution and joining two fixed points. This survey is devoted to the singular curves, which can be length minimizing geodesics, independent of the choice of inner product.     

Output Controllability for Semi-Linear Distributed Parabolic Systems

The purpose of this paper is to explore regional controllability developed for linear system [5, 15] to the semi-linear case. Under the hypothesis that the approximate regional controllability holds, we find a control which steers a system to a desired state in a subregion of the system domain and the approach is based on an extension of the Hilbert uniqueness method and the Schauder fixed-point theorem. The analytical case is then discussed using generalized inverse techniques and successfully illustrated by simulations.      

Topological obstructions to the representability of functions by quadratures

A topological variant of Galois theory, in which the monodromy group plays the role of the Galois group, is described. It turns out that there are topological restrictions on the way the Riemann surface of a function represented by quadratures covers the complex plane.This work was partially supported by Grant MBF000 from the International Science Foundation.     

On the confluence phenomenon for fuchsian equations

In the paper we present a method of investigation of asymptotic behavior of solutions to parameter-dependent degenerate differential equations both with regular and irregular points of singularity. We examine the confluence phenomenon for such points, which is the process of the coincidence of different points at a certain value of the parameter. This examination is based on a resurgent representation of the corresponding solutions which also depends on the parameter. In particular, the confluence of the representations in question is considered.     

On problems related to growth, entropy, and spectrum in group theory

We review some known results and open problems related to the growth of groups. For a finitely generated group Γ, given whenever necessarytogether with a finite generating set, we discuss the notions of

Approximate Invariance and Differential Inclusions in Hilbert Spaces

Consider a mappingF from a Hilbert spaceH to the subsets ofH, which is upper semicontinuous/Lipschitz, has nonconvex, noncompact values, and satisfies a linear growth condition. We give the first necessary and sufficient conditions in this general setting for a subsetS ofH to be approximately weakly/strongly invariant with respect to approximate solutions of the differential inclusion . The conditions are given in terms of the lower/upper Hamiltonians corresponding toF and involve nonsmooth analysis elements and techniques. The concept of approximate invariance generalizes the well-known concept of invariance and in turn relies on the notion of an ∈-trajectory corresponding to a differential inclusion. Dedicated to Professor R. V. Gamkrelidze The first and the third authors are supported in part by the Natural Sciences and Engineering Research Council of Canada and Le Fonds FCAR du Québec. The second author is visiting Rutgers University. Supported in part by the Russian Foundation for Fundamental Research, Grant 96-01-00219, by the Natural Sciences and Engineering Research Council of Canada, Le Fonds FCAR du Québec, and by the Rutgers Center for Systems and Control (SYCON).     

Nonabelian statistics for calogero-sutherland systems of particles

The mathematical aspects of the theory of nonabelian statistics and anyons are discussed. The dynamics and kinematics of these objects are defined by Hamiltonians which are the sums of terms associated with different roots of a given root system. The equations of Fuchsian type which are the so-called generalized Knizhnik-Zamolodchikov equations associated with the root systemsA _n ,B _n , andC _n , are presented. Some results the generalized pure braid groups and their application for defining the generalized anyon are discussed. The factorization of the spin Calogero-Sutherland operators and their connection with the theory of nonabelian statistics and anyons are given. To Dmitrii Anosov with admiration and deep respect on his sixtieth birthday The research was supported by Grants RFFI-INTAS 00418 and RFFI-Germany 96-01-00008G.     

Parametric Stability of Impulsive Functional Differential Equations

The purpose of this paper is to obtain sufficient conditions for parametric stability of impulsive functional differential equations with impulse perturbations at fixed moments of time. The concept of parametric stability provides a mathematical framework for solving the joint problem of feasibility and stability of equilibrium states. This problem arises whenever uncertain parameters can cause the equilibrium states to shift or disappear. The main results are obtained by means of piecewise continuous functions, which are analogs of the classical Lyapunov functions, and via the Razumikhin technique. Two examples are given to illustrate our results.      

Generalized Rolle theorem in ℝ n and ℂ

The Rolle theorem for functions of one real variable asserts that the number of zeros off on a real connected interval can be at most that off plus 1. The following inequality is a multidimensional generalization of the Rolle theorem: if [0,1] ⁿ ,tx(t), is a closed smooth spatial curve and L() is the length of its spherical projection on a unit sphere, then for thederived curve [0,1], ⁿ , the following inequality holds: L() L(). For the analytic functionF(z) defined in a neighborhood of a closed plane curve ² this inequality implies that (F) (F) + (), where (F) is the total variation of the argument ofF along , and () is the integral absolute curvature of .As an application of this inequality, we find an upper bound for the number of complex isolated zeros of quasipolynomials. We also establish a two-sided inequality between the variation index (F) and another quantity, called theBernstein index, which is expressed in terms of the modulus growth of an analytic function.     

Finite singularities of nonlinear systems. Output stabilization,observability, and observers

In this paper, for the purposes, first, of constructing nonlinear observers and, second, of output stabilization for observed nonlinear systems, we develop a theory allowing one to deal with singularities that can appear. In the uncontrolled analytic case, we are especially interested in finite singularities.Using this theory, we generalize some of our previous results on the construction of nonlinear observers.     

On Implicit Second-Order Ordinary Differential Equations: Completely Integrable and Clairaut Type

In this paper, we give a characterization of implicit second-order ordinary differential equations with smooth complete integrals which we call Clairaut-type equations. Moreover, we consider properties of the Clairaut-type equations and present the duality among special completely integrable equations with respect to Engel–Legendre transformations.      

Well-posedness and regularity of weakly coupled wave-plate equation with boundary control and observation

We consider the open-loop system of a weakly coupled linear wave-plate equation with Dirichlet boundary control and collocated observation. It is shown that the system is well-posed in the sense of D. Salamon and regular in the sense of G. Weiss. With the multiplier method, the feedthrough operator is explicitly represented. This work was supported by the National Natural Science Foundation of China (No.60774014), Program for the Top Young Academic Leaders of Higher Learning Institutions of Shanxi and the National Research Foundation of South Africa.     

Flows on closed surfaces and behavior of trajectories lifted to the universal covering plane

A survey of works concerning a kind of generalization of the Poincaré rotation number for a rather general class of flows on closed surfaces is given. The idea (going back to A. Weil) is to replace this number by the asymptotic direction at infinity of the trajectory lifted to the universal covering plane. Other works on flows on surfaces are also mentioned, but this is done primarily for comparison in order to shed light on the subject.The work was partially supported by grant 93-011-1727 of the Russian foundation for fundamental research, by a research grant from the American Mathematical Society, by the Soros stipend of the ISF — Academy of Natural Sciences (Russia), and by grant M1E000 of the International Science Foundation.     

Rigid and abnormal line subdistributions of 2-distributions

A line subdistributionL of the distributionE on then-manifoldM is said to be rigid (abnormal) if anyL-curve is rigid (abnormal), i.e., is an isolated (singular) point of the set of allE-curves joining two fixed points ofM. This paper contains the necessary and sufficient conditions for a line subdistribution to be rigid (abnormal), formulated in geometrical and algebraic terms, the existence theorems, and the analysis of the structure of rigid (abnormal), line subdistributions of generic 2-disrtibutions.Research partly supported by the Technion VRP Fund.     

Homogeneous tangent vectors and high-order necessary conditions for optimal controls

An extension of the classical Pontryagin maximum principle for Mayer problems without terminal constraints, subject to affine control systems , is proved. In connection with a suitable dilation on the state space ℝ ⁿ , we introduce a class of “homogeneous tangent vectors” which provide a nonlinear, high-order approximation of the attainable set in the case where the usual linear approximation proves to be inadequate. By studying control variations which generate homogeneous tangent vectors, we derive new necessary conditions for optimality that are particularly effective for basically nonlinear optimal control problems where other high-order tests provide no conclusive information. This research was partially supported by Istituto Nazionale di Alta Matematica “F. Severi.”